Jet Energy and Other Parameters for the Afterglows of GRB 980703, GRB 990123, GRB 990510, and GRB 991216 Determined from Modeling of Multifrequency Data

Abstract

We model the radio, optical, and X-ray emission for the afterglows of GRB 980703, GRB 990123, GRB 990510, and GRB 991216, within the framework of relativistic jets, to determine their physical parameters. The models that yield acceptable Ðts to the data have jet energies mostly between 1050 and 1051 ergs and initial opening angles between 1¡ and 4¡. The external medium density is uncertain by at least 1 order of magnitude in each case, being around 10~3 cm~3 for GRB 980703 and GRB 990123, D10~1 cm~3 for GRB 990510, and D3 cm~3 for GRB 991216. If the jets are uniform (i.e., there are no angular gradients of the energy per solid angle), then the 20 keVÈ1 MeV radiative efficiency during the GRB phase must have been at least 2%È3% for GRB 990510, 20% for GRB 990123, and 30% for GRB 991216. Subject headings : gamma rays : bursts È ISM : jets and outÑows È methods : numerical È radiation mechanisms : nonthermal È shock waves